# Gauss Test for Convergence

1. Oct 5, 2011

### process91

1. The problem statement, all variables and given/known data

I just got done proving Gauss' test, which is given in the book as:

If there is an $N\ge 1$, an $s>1$, and an $M>0$ such that
$$\frac{a_{n+1}}{a_n}=1 - \frac{A}{n} + \frac{f(n)}{n^s}$$
where $|f(n)|\le M$ for all n, then $\sum a_n$ converges if $A>1$ and diverges if $A \le 1$.

This is equivalent to the many other forms I have found on the web. The next question asks to use this test to prove that the series

$$\sum_{n=1} ^{\infty} \left( \frac{1 \cdot 3 \cdot 5 \cdot \cdot \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot \cdot \cdot (2n)} \right)^k$$

converges if $k>2$ and diverges if $k \le 2$ using Gauss' test.

2. Relevant equations

3. The attempt at a solution

OK, so much for the preamble. Here's my attempt:

$$\frac{a_{n+1}}{a_n} = \left ( \frac {2n+1}{2n+2} \right ) ^ k$$

And that's it... I don't know how to put this into a form which corresponds in general to the form required for Gauss' test. I saw a similar problem online solved with the use of the fact that $\left (\frac{n}{n+1} \right)^k \approx 1 - \frac{k}{n}$ for large n, but the book has not covered anything like that (it introduced the ~ symbol, but not $\approx$).